# Expression of an abelian group

Let $A$ be a abelian group generated by elements $\langle a_1,a_2,a_3\rangle$ and $B$ be a subgroup generated by $\langle b_1,b_2,b_3\rangle$ where $\begin{pmatrix} b_1\\ b_2\\b_3 \end{pmatrix}= \begin{pmatrix} a_1&a_2&0\\ a_1&0&a_3\\ 0&a_2&a_3 \end{pmatrix}\begin{pmatrix}\alpha\\ \beta\\ \gamma\end{pmatrix}$ and $\alpha,\beta, \gamma \in \mathbb Z$

How then might we express $A/B$ as $\bigoplus_i \mathbb Z_{m_i}$?

• sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf – Dylan Moreland Mar 17 '12 at 18:18
• @DylanMoreland: Is the smith form always achievable? I am not so sure... – alex b Mar 17 '12 at 20:39
• @DylanMoreland: Say $\alpha=5,\beta=3,\gamma=10$? – alex b Mar 17 '12 at 21:04
• @alexb: Yes, the Smith Normal Form is always achievable. – Arturo Magidin Mar 17 '12 at 21:11
• @Arturo, I hope you find that 5 that you lost in your head - you might need it some day. – Gerry Myerson Mar 19 '12 at 5:53

• Thank you, Rolando, but I am not sure tht the SNF is always achievable? Say $\alpha=5,\beta=3,\gamma=10$? – alex b Mar 17 '12 at 21:06